arXiv subjects

Erik Orvehed Hiltunen

Publications and source records attributed to Erik Orvehed Hiltunen.

At least 19 recordsLinked to original sources

arXiv2026

Limit cycles govern wave energy accumulation in space-time-varying materials

We study the amplification and concentration of wave energy in smooth, periodic space-time media through the dynamics of the characteristic curves of the associated wave equation. In an impedance-matched setting, we show that stable limit cycles attract neighbouring characteristics, focusing the wave energy density. The exponential attraction rate is estimated analytically from the system's Lyapunov exponent. We further show that the attraction of characteristics and energy accumulation persist under a range of nonperiodic perturbations. Finally, we present a Floquet-Fourier numerical method for computing stable and unstable limit cycles and their attraction rates. Throughout, we exemplify the results using the example of modulation given by a superposition of travelling waves, revealing saddle--node bifurcations where limit cycles may appear depending on the relative wave speeds. Full-wave simulations confirm that the characteristic mechanism continues to describe energy accumulation beyond the impedance-matched case.

math.AP↗

arXiv2026

Non-Bloch band theory for finite banded non-Hermitian Toeplitz matrices

Non-Bloch band theory recovers the open-boundary spectrum of a non-Hermitian chain from its Bloch Hamiltonian evaluated along the generalised Brillouin zone rather than the unit circle. We develop its finite-size counterpart for banded Toeplitz matrices with real open-boundary spectrum. Reality of the spectrum forces the generalised Brillouin zone to be parametrised by an analytic polar curve, so the known sufficient condition for a real spectrum is also necessary under some mild non-degeneracy conditions. At finite size $n$, the eigenvalues of the original and the symmetrised Bloch Hamiltonians are given by samples at $n$ quantised momenta from the same limiting band, although separated by a phase mismatch of order $1/n$ which we obtain in closed form. Iterating this correction $K$ times yields a quantisation rule on the generalised Brillouin zone reproducing the finite spectrum with an error $\mathcal{O}(n^{-K-1})$ for any fixed $K$. At finite size, the generalised Brillouin zone itself is unchanged and only its parametrisation depends on $n$.

math.SP↗

arXiv2026

Mathematical Foundation for the Generalised Brillouin zone of m-banded Toeplitz operators

We show that the spectrum of the open-boundary limit of banded Toeplitz matrices is real whenever the associated symbol function is real-valued along a closed polar curve. Building on this result, we develop both analytical and numerical methods to symmetrise a class of banded non-Hermitian Toeplitz matrices whose asymptotic spectra are real. Finally, we provide a rigorous mathematical foundation for the generalised Brillouin zone, a concept widely used in non-Hermitian physics, by proving that it coincides with the polar curve on which the symbol function takes real values.

math.SP↗

arXiv2026

Topologically protected interface modes in multi-band damped lattice models

Tridiagonal $k$-Toeplitz operators provide a natural framework for modelling one-dimensional $k$-periodic lattice systems. A fundamental connection is obtained between Coburn's lemma for tridiagonal $k$-Toeplitz operators and the existence of edge modes. We reveal that topological edge modes are characterised by the eigenvalues of the leading principal submatrix of the symbol function. A complete analysis of tridiagonal interface operators satisfying global inversion symmetry is then presented. These results are applied to finite one-dimensional $k$-periodic chains of damped resonators that satisfy both local and global inversion symmetry. Additionally, disordered tight-binding interface operators are shown to support a topologically robust zero-energy interface state. Numerical simulations are conducted to illustrate the theoretical findings.

math.SP↗

arXiv2026

Non-Hermitian Fabry-Pérot Resonances

We characterise non-Hermitian Fabry-Pérot resonances in high-contrast resonator systems and study the properties of their associated resonant modes from continuous differential models. We consider two non-Hermitian effects: the exceptional point degeneracy and the skin effect induced by imaginary gauge potentials. Using the propagation matrix formalism, we characterise these two non-Hermitian effects beyond the subwavelength regime. This analysis allows us to (i) establish the existence of exceptional points purely from radiation conditions and to (ii) prove that the non-Hermitian skin effect applies uniformly across resonant modes, yielding broadband edge localisation.

math-ph↗

arXiv2026

A Tight-binding Approach for Computing Subwavelength Guided Modes in Crystals with Line Defects

In this paper, we develop an accurate and efficient framework for computing subwavelength guided modes in high-contrast periodic media with line defects, based on a tight-binding approximation. The physical problem is formulated as an eigenvalue problem for the Helmholtz equation with high-contrast parameters. By employing layer potential theory on unbounded domains, we characterize the subwavelength frequencies via the quasi-periodic capacitance matrix. Our main contribution is the proof of exponential decay of the off-diagonal elements of the associated full and quasi-periodic capacitance matrices. These decay properties provide error bounds for the banded approximation of the capacitance matrices, thereby enabling a tight-binding approach for computing the spectral properties of subwavelength resonators with non-compact defects. Various numerical experiments are presented to validate the theoretical results, including applications to topological interface modes.

math-ph↗

arXiv2025

Spectra and pseudospectra of non-Hermitian Toeplitz operators: Eigenvector decay transitions in banded and dense matrices

Using a generalised Floquet-Bloch theory, we present a mathematical method to construct eigenvectors for non-Hermitian Toeplitz operators. We extend the method to both banded Toeplitz operators and those with algebraically decaying, fully dense off-diagonal structure. We present sharp decay estimates for the amplitude of bulk eigenmodes as well as eigenmodes associated with defect eigenfrequencies inside the spectral band gap. The validity of those results is illustrated numerically and we show that banded approximations give poor reconstructions of the dense operators, due to the slow algebraic decay. We apply the insights gained to model the non-Hermitian skin effect in a three-dimensional system of subwavelength resonators, where the corresponding operator exhibits only algebraic decay of off-diagonal entries. We use our approach to demonstrate the fundamental mechanism responsible for the transition between the non-Hermitian skin effect and defect-induced localisation in the bulk.

math.AP↗

arXiv2025

Optical theorem and generalized energy conservation for scattering of time-modulated waves

We introduce and study a generalized energy conservation relation for scattering of time-modulated waves, where conventional energy conservation does not hold. Based on this relation, we derive an optical theorem and compute the active power describing the power input or output due to scattering. Notably, the same system may be subject to energy gain, energy loss, or energy conservation depending on the frequency harmonics present in the wave field. Moreover, we show how the optical theorem derived herein may be used for image reconstruction based on measurements of the power. In particular, such measurements do not require information about the phase of the scattered field.

physics.optics↗

arXiv2025

Effective Medium Theory for Time-modulated Subwavelength Resonators

This paper provides a general framework for deriving effective material properties of one-dimensional, time-modulated systems of subwavelength resonators. It applies to subwavelength resonator systems with a general form of time-dependent parameters. We show that the resonators can be accurately described by a point-scattering formulation when the width of the resonators is small. In contrast to the static setting, where this point interaction approximation yields a Lippmann-Schwinger equation for the effective material properties, the mode coupling in the time-modulated case instead yields an infinite linear system of Lippmann-Schwinger-type equations. The effective equations can equivalently be written as a system of differential equations. Moreover, we introduce a numerical scheme to approximately solve the system of coupled equations and illustrate the validity of the effective equation.

physics.optics↗

arXiv2025

Energy Balance and Optical Theorem for Time-Modulated Subwavelength Resonator Arrays

We study wave propagation through a one-dimensional array of subwavelength resonators with periodically time-modulated material parameters. Focusing on a high-contrast regime, we use a scattering framework based on Fourier expansions and scattering matrix techniques to capture the interactions between an incident wave and the temporally varying system. This way, we derive a formulation of the total energy flux corresponding to time-dependent systems of resonators. We show that the total energy flux is composed of the transmitted and reflected energy fluxes, and derive an optical theorem which characterises the energy balance of the system. We provide a number of numerical experiments to investigate the impact of the time-dependency, the operating frequency and the number of resonators on the maximal attainable energy gain and energy loss. Moreover, we show the existence of lasing points, at which the total energy diverges. Our results lay the foundation for the design of energy dissipative or energy amplifying systems.

physics.optics↗

arXiv2025

Complex Band Structure and localisation transition for tridiagonal non-Hermitian k-Toeplitz operators with defects

Using the Bloch-Floquet theory, we propose an innovative technique to obtain the eigenvectors of tridiagonal k-Toeplitz operators. This method offers a more extensive and quantitative basis for describing localised eigenvectors beyond the non-trivial winding zone, yielding sharp decay bounds. The validity of our results is confirmed numerically in one-dimensional resonator chains, showcasing non-Hermitian skin localisation, bulk localisation, and tunnelling effects. We conclude the paper by analysing non-Hermitian tight binding Hamiltonians, illustrating the broad applicability of the complex band structure.

math.AP↗

arXiv2025

Universal estimates for the density of states for aperiodic block subwavelength resonator systems

We consider the spectral properties of aperiodic block subwavelength resonator systems in one dimension, with a primary focus on the density of states. We prove that for random block configurations, as the number of blocks $M\to \infty$, the integrated density of states converges to a non-random, continuous function. We show both analytically and numerically that the density of states exhibits a tripartite decomposition: it vanishes identically within bandgaps; it forms smooth, band-like distributions in shared pass bands (a consequence of constructive eigenmode interactions); and, most notably, it exhibits a distinct fractal-like character in hybridisation regions. We demonstrate that this fractal-like behaviour stems from the limited interaction between eigenmodes within these hybridisation regions. Capitalising on this insight, we introduce an efficient meta-atom approach that enables rapid and accurate prediction of the density of states in these hybridisation regions. This approach is shown to extend to systems with quasiperiodic and hyperuniform arrangements of blocks.

math-ph↗

arXiv2025

Complex Brillouin Zone for Localised Modes in Hermitian and Non-Hermitian Problems

We develop a mathematical and numerical framework for studying evanescent waves in subwavelength band gap materials. By establishing a link between the complex Brillouin zone and various Hermitian and non-Hermitian phenomena, including defect localisation in band gap materials and the non-Hermitian skin effect, we provide a unified perspective on these systems. In two-dimensional structures, we develop analytical techniques and numerical methods to study singularities of the complex band structure. This way, we demonstrate that gap functions effectively predict the decay rates of defect states. Furthermore, we present an analysis of the Floquet transform with respect to complex quasimomenta. Based on this, we show that evanescent waves may undergo a phase transition, where local oscillations drastically depend on the location of corresponding frequency inside the band gap.

math.AP↗

arXiv2025

Perturbation theory for dispersion relations of spacetime-periodic materials

We consider Bloch states of weak spacetime-periodic perturbations of homogeneous materials in one spatial dimension. The interplay of space- and time-periodicity leads to an infinitely degenerate dispersion relation in the free case. We consider a general perturbation term, and, as consequence of the infinite degeneracy, we show that the effective equations are given by the eigenvalue problem of an infinite matrix. Our method can be viewed as a time-modulated generalisation of the nearly-free electron model. Based on this result, we find that the infinite degeneracy may split into a family of non-degenerate bands. Our results are illustrated with numerical calculations, and we observe close agreement between the perturbation theory and the numerically computed full solution.

cond-mat.mes-hall↗

arXiv2025

Space-Time Wave Localisation in Systems of Subwavelength Resonators

In this paper we study the dynamics of metamaterials composed of high-contrast subwavelength resonators and show the existence of localised modes in such a setting. A crucial assumption in this paper is time-modulated material parameters. We prove a so-called capacitance matrix approximation of the wave equation in the form of an ordinary differential equation. These formulas set the ground for the derivation of a first-principles characterisation of localised modes in terms of the generalised capacitance matrix. Furthermore, we provide numerical results supporting our analytical results showing for the first time the phenomenon of space-time localised waves in a perturbed time-modulated metamaterial. Such spatio-temporal localisation is only possible in the presence of subwavelength resonances in the unperturbed structure. We introduce the time-dependent degree of localisation to quantitatively determine the localised modes and provide a variety of numerical experiments to illustrate our formulations and results.

math.AP↗

arXiv2024

Topological interface modes in systems with damping

We extend the theory of topological localised interface modes to systems with damping. The spectral problem is formulated as a root-finding problem for the interface impedance function and Rouché's theorem is used to track the zeros when damping is introduced. We show that the localised eigenfrequencies, corresponding to interface modes, remain for non-zero dampings. Using the transfer matrix method, we explicitly characterise the decay rate of the interface mode.

math.AP↗

arXiv2024

Complex Band Structure for Subwavelength Evanescent Waves

We present the mathematical and numerical theory for evanescent waves in subwavelength band gap materials. We begin in the one-dimensional case, whereby fully explicit formulas for the complex band structure, in terms of the capacitance matrix, are available. As an example, we show that the gap functions can be used to accurately predict the decay rate of the interface mode of a photonic analogue of the SSH-model. In two dimensions, we derive the band gap Green's function and characterise the subwavelength gap functions via layer potential techniques. By generalising existing lattice-summation techniques, we illustrate our results numerically by computing the complex band structure in a variety of settings.

math.AP↗

arXiv2024

Coupled harmonics due to time-modulated point scatterers

We consider the resonance and scattering properties of a composite medium containing scatterers whose properties are modulated in time. When excited with an incident wave of a single frequency, the scattered field consists of a family of coupled harmonics at frequencies differing by the frequency of temporal modulation. Similarly, the temporal modulation induces coupling between the resonance frequencies, leading to exceptional points at certain modulation amplitudes. Moreover, the lack of energy conservation causes scattering coefficients to blow up when (complex) resonances cross the real axis. We have developed an integral operator approach to characterize the scattering problem and, for high-contrast scatterers, we present small-volume asymptotic formulas analogous to the classical results for the static (unmodulated) case. We conclude the paper with a boundary integral formulation of the time-modulated problem, which gives an efficient numerical approach and corroborates the asymptotic formulas.

cond-mat.mes-hall↗